March  2020, 9(1): 1-25. doi: 10.3934/eect.2020014

Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints

Laboratory of Functional Analysis and Geometry of Spaces, Faculty of Mathematics and Computer Sciences, University of M'sila, 28000, Algeria

Received  November 2017 Revised  May 2018 Published  March 2020 Early access  October 2019

Fund Project: The author is supported by the Algerian Ministry of Higher Education and Scientific Research, under a PRFU Project No. C00L03UN280120180007.

We study the wave equation in an interval with two linearly moving endpoints. We give the exact solution by a series formula, then we show that the energy of the solution decays at the rate $ 1/t $. We also establish observability results, at one or at both endpoints, in a sharp time. Moreover, using the Hilbert uniqueness method, we derive exact boundary controllability results.

Citation: Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations and Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014
References:
[1]

N. L. Balazs, On the solution of the wave equation with moving boundaries, J. Math. Anal. Appl., 3 (1961), 472-484.  doi: 10.1016/0022-247X(61)90071-3.

[2]

C. Bardos and G. Chen, Control and stabilization for the wave equation. Ⅲ: Domain with moving boundary, SIAM J. Control Optim., 19 (1981), 123-138.  doi: 10.1137/0319010.

[3]

G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, 4$^{th}$ edition, John Wiley & Sons, Inc., New York, 1989.

[4]

J. Cooper and C. Bardos, A nonlinear wave equation in a time dependent domain, J. Math. Anal. Appl., 42 (1973), 29-60.  doi: 10.1016/0022-247X(73)90120-0.

[5]

L. Z. CuiX. Liu and H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, J. Math. Anal. Appl., 402 (2013), 612-625.  doi: 10.1016/j.jmaa.2013.01.062.

[6]

V. V. DodonovA. B. Klimov and D. E. Nikonov, Quantum phenomena in resonators with moving walls, J. Math. Phys., 34 (1993), 2742-2756.  doi: 10.1063/1.530093.

[7]

E. Knobloch and R. Krechetnikov, Problems on time-varying domains: Formulation, dynamics, and challenges, Acta Appl. Math., 137 (2015), 123-157.  doi: 10.1007/s10440-014-9993-x.

[8]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.

[9]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2005. doi: 10.1007/b139040.

[10]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod-Gautier Villars, Paris, 1969.

[11]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systemes Distribués. Tome 1. Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, 8. Masson, Paris, 1988.

[12]

L. Q. LuS. J. LiG. Chen and P. F. Yao, Control and stabilization for the wave equation with variable coefficients in domains with moving boundary, Systems & Control Lett., 80 (2015), 30-41.  doi: 10.1016/j.sysconle.2015.04.003.

[13]

L. A. MedeirosJ. Limaco and S. B. Menezes, Vibrations of elastic strings: Mathematical aspects, part two, J. Comput. Anal. Appl., 4 (2002), 211-263.  doi: 10.1023/A:1013151525487.

[14]

M. Milla Miranda, Exact controllability for the wave equation in domains with variable boundary, Rev. Mat. Complut. Madrid, 9 (1996), 435-457.  doi: 10.5209/rev_rema.1996.v9.n2.17595.

[15]

G. T. Moore, Quantum theory of electromagnetic field in a variable-length one-dimensional cavity, J. Math. Phys., 11 (1970), 2679-2691.  doi: 10.1063/1.1665432.

[16]

M. A. Pinsky, Introduction to Fourier Analysis and Wavelets, Graduate Studies in Mathematics, 102. American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/102.

[17]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.

[18]

A. Sengouga, Observability and controllability of the 1-D wave equation in domains with moving boundary, Acta Appli. Math., 157 (2018), 117-128.  doi: 10.1007/s10440-018-0166-1.

[19]

A. Sengouga, Observability of the 1-D wave equation with mixed boundary conditions in a non-cylindrical domain, Mediterr. J. Math., 15 (2018), Art. 62, 22 pp. doi: 10.1007/s00009-018-1107-y.

[20]

H. C. Sun, H. F. Li and L. Q. Lu, Exact controllability for a string equation in domains with moving boundary in one dimension, Electron. J. Diff. Equations, 2015 (2015), 7 pp.

[21]

P. K. C. Wang, Stabilization and control of distributed systems with time-dependent spatial domains, J. Optim. Theory Appl., 65 (1990), 331-362.  doi: 10.1007/BF01102351.

[22]

E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev., 47 (2005), 197-243.  doi: 10.1137/S0036144503432862.

show all references

References:
[1]

N. L. Balazs, On the solution of the wave equation with moving boundaries, J. Math. Anal. Appl., 3 (1961), 472-484.  doi: 10.1016/0022-247X(61)90071-3.

[2]

C. Bardos and G. Chen, Control and stabilization for the wave equation. Ⅲ: Domain with moving boundary, SIAM J. Control Optim., 19 (1981), 123-138.  doi: 10.1137/0319010.

[3]

G. Birkhoff and G.-C. Rota, Ordinary Differential Equations, 4$^{th}$ edition, John Wiley & Sons, Inc., New York, 1989.

[4]

J. Cooper and C. Bardos, A nonlinear wave equation in a time dependent domain, J. Math. Anal. Appl., 42 (1973), 29-60.  doi: 10.1016/0022-247X(73)90120-0.

[5]

L. Z. CuiX. Liu and H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, J. Math. Anal. Appl., 402 (2013), 612-625.  doi: 10.1016/j.jmaa.2013.01.062.

[6]

V. V. DodonovA. B. Klimov and D. E. Nikonov, Quantum phenomena in resonators with moving walls, J. Math. Phys., 34 (1993), 2742-2756.  doi: 10.1063/1.530093.

[7]

E. Knobloch and R. Krechetnikov, Problems on time-varying domains: Formulation, dynamics, and challenges, Acta Appl. Math., 137 (2015), 123-157.  doi: 10.1007/s10440-014-9993-x.

[8]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, Masson, Paris, John Wiley & Sons, Ltd., Chichester, 1994.

[9]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics, Springer-Verlag, New York, 2005. doi: 10.1007/b139040.

[10]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod-Gautier Villars, Paris, 1969.

[11]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systemes Distribués. Tome 1. Contrôlabilité Exacte, Recherches en Mathématiques Appliquées, 8. Masson, Paris, 1988.

[12]

L. Q. LuS. J. LiG. Chen and P. F. Yao, Control and stabilization for the wave equation with variable coefficients in domains with moving boundary, Systems & Control Lett., 80 (2015), 30-41.  doi: 10.1016/j.sysconle.2015.04.003.

[13]

L. A. MedeirosJ. Limaco and S. B. Menezes, Vibrations of elastic strings: Mathematical aspects, part two, J. Comput. Anal. Appl., 4 (2002), 211-263.  doi: 10.1023/A:1013151525487.

[14]

M. Milla Miranda, Exact controllability for the wave equation in domains with variable boundary, Rev. Mat. Complut. Madrid, 9 (1996), 435-457.  doi: 10.5209/rev_rema.1996.v9.n2.17595.

[15]

G. T. Moore, Quantum theory of electromagnetic field in a variable-length one-dimensional cavity, J. Math. Phys., 11 (1970), 2679-2691.  doi: 10.1063/1.1665432.

[16]

M. A. Pinsky, Introduction to Fourier Analysis and Wavelets, Graduate Studies in Mathematics, 102. American Mathematical Society, Providence, RI, 2009. doi: 10.1090/gsm/102.

[17]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.

[18]

A. Sengouga, Observability and controllability of the 1-D wave equation in domains with moving boundary, Acta Appli. Math., 157 (2018), 117-128.  doi: 10.1007/s10440-018-0166-1.

[19]

A. Sengouga, Observability of the 1-D wave equation with mixed boundary conditions in a non-cylindrical domain, Mediterr. J. Math., 15 (2018), Art. 62, 22 pp. doi: 10.1007/s00009-018-1107-y.

[20]

H. C. Sun, H. F. Li and L. Q. Lu, Exact controllability for a string equation in domains with moving boundary in one dimension, Electron. J. Diff. Equations, 2015 (2015), 7 pp.

[21]

P. K. C. Wang, Stabilization and control of distributed systems with time-dependent spatial domains, J. Optim. Theory Appl., 65 (1990), 331-362.  doi: 10.1007/BF01102351.

[22]

E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods, SIAM Rev., 47 (2005), 197-243.  doi: 10.1137/S0036144503432862.

Figure 1.  Extension of an initial data $ \phi ^{0} $ when $ v _{1}<v _{2} $
Figure 2.  Propagation of a wave with a small support near an endpoint $ (v _{2}<v _{1}) $
Figure 3.  Propagation of small disturbances with supports near one or two ends $ (v _{1}<v _{2}) $
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